lanman wrote:

-snip-
The mortar is disintegrating, and if not corrected, would
presumably lead to CO leaks or in the worst case - collapse.
-snip-

Collapse is expensive, but probably wouldn't hurt anyone. CO
poisoning is silent and deadly. Before you do anything buy a couple
CO detectors and put one on each floor of your house.

And shop around for another chimney guy. I can't see it from here,
but it is unlikely that your flue can't be lined.

Jim